Control Systems Laboratory Manual

DIGITAL SIMULATION OF LINEAR SYSTEM EXPT.NO : DATE AIM: To simulate the time response characteristic of higher-order Multiinput multi output (MIMO) liner system using state variable formulation. APPARATUS REQUIRED: MATLAB 6.5 THEORY: Time Domain Specification The desired performance characteristics of control systems are specified in terms of time domain specification. System with energy storage elements cannot respond instantaneously and will exhibit transient responses, whenever they are subjected to inputs or disturbances. The desired performance characteristics of a system of any order may be specified in terms of the transient response to a units step input signal. The transient response of a system to a unit step input depends on the initial conditions. Therefore to compare the time response of various systems it is necessary to start with standard initial conditions. The most practical standard is to start with the system at rest and output and all time derivatives there of zero. The transient response of a practical control system often exhibits damped oscillation before reaching steady state. The transient response characteristics of a control system to a unit step input are specified in terms of the following time domain specifications. :

PROCEDURE: 7. Enter the command window of the MATLAB. 8. Create a new workspace by selecting new file. 9. Complete your model. 10.Run the model by either pressing F5 or start simulation. 11.View the results. 12.Analysis the stability of the system for various values of gain. PROBLEM: Obtain the step response of series RLC circuit with R = 1.3KΩ , L = 26mH and C=3.3µ f using MATLAB M – File.

MATLAB PROGRAM FOR UNIT IMPULSE PRSPONSE:
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STABILITY ANALYSIS OF LINEAR SYSTEM
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Control Systems Laboratory Manual

EXPT.NO : DATE AIM:
(i)

:

To obtain the bode plot, Nyquist plot and root locus of the given transfer function. To analysis the stability of given linear system using MATLAB.

(ii)

APPARATUS REQUIRED: System with MATLAB THEORY: Frequency Response: The frequency response is the steady state response of a system when the input to the system is a sinusoidal signal. Frequency response analysis of control system can be carried either analytically or graphically. The various graphical techniques available for frequency response analysis are 1. Bode Plot 2. Polar plot (Nyquist plot) 3. Nichols plot 4. M and N circles 5. Nichols chart Bode plot: The bode plot is a frequency response plot of the transfer function of a system. A bode plot consists of two graphs. One is plot of the magnitude of a sinusoidal transfer function versus log ω . The other is plot of the phase angle of a sinusoidal transfer function versus logω . The main advantage of the bode plot is that multiplication of magnitude can be converted into addition. Also a simple method for sketching an approximate log magnitude curve is available.
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Control Systems Laboratory Manual

Polar plot: The polar plot of a sinusoidal transfer function G (jω ) on polar coordinates as ω is varied from zero to infinity. Thus the polar plot is the locus of vectors [G (jω ) ] ∠ G (jω ) as ω is varied from zero to infinity. The polar plot is also called Nyquist plot. Nyquist Stability Criterion: If G(s)H(s) contour in the G(s)H(s) plane corresponding to Nyquist contour in s-plane encircles the point – 1+j0 in the anti – clockwise direction as many times as the number of right half s-plain of G(s)H(s). Then the closed loop system is stable. Root Locus: The root locus technique is a powerful tool for adjusting the location of closed loop poles to achieve the desired system performance by varying one or more system parameters. The path taken by the roots of the characteristics equation when open loop gain K is varied from 0 to ∞ are called root loci (or the path taken by a root of characteristic equation when open loop gain K is varied from 0 to ∞ is called root locus.) Frequency Domain Specifications: The performance and characteristics of a system in frequency domain are measured in term of frequency domain specifications. The requirements of a system to be designed are usually specified in terms of these specifications. The frequency domain specifications are 1. Resonant peak, Mr 2. Resonant Frequency, ω r. 3. Bandwidth. 4. Cut – off rate 5. Gain margin 6. Phase margin

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Control Systems Laboratory Manual

Resonant Peak, Mr The maximum value of the magnitude of closed loop transfer function is called the resonant peak, Mr. A large resonant peak corresponds to a large over shoot in transient response. Resonant Frequency, ω
r

The bandwidth is the range of frequency for which the system gain is more than -3db. The frequency at which the gain is -3db is called cut off frequency. Bandwidth is usually defined for closed loop system and it transmits the signals whose frequencies are less than cut-off frequency. The bandwidth is a measured of the ability of a feedback system to produce the input signal, noise rejection characteristics and rise time. A large bandwidth corresponds to a small rise time or fast response. Cut-Off Rate: The slope of the log-magnitude curve near the cut off frequency is called cut-off rate. The cut-off rate indicates the ability of the system to distinguish the signal from noise. Gain Margin, Kg The gain margin, Kg is defined as the reciprocal of the magnitude of open loop transfer function at phase cross over frequency. The frequency at witch the phase of open loop transfer function is 180 is called the phase cross over frequency, ω pc. Phase Margin, γ The phase marginγ , is that amount of additional phase lag at the gain cross over frequency required to bring the system to the verge of instability, the gain cross over frequency ω gc is the frequency at which the magnitude of open loop transfer function is unity (or it is the frequency at which the db magnitude is zero). PROCEDURE:

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Control Systems Laboratory Manual

1. 2. 3. 4. 5. 6.

Enter the command window of the MATLAB. Create a new M – file by selecting File – New – M – File. Type and save the program. Execute the program by either pressing F5 or Debug – Run. View the results. Analysis the stability of the system for various values of gain.

AIM: To find the percentage peak overshoot and steady state error of the given P, PI and PID APPARATUS REQUIRED: MATLAB, SIMULINK THEORY: A controller is similar to an amplifier. It is used in closed loop control system to enhance the system output. In proportional controller, the output is proportional to the input. Kp represents the gain constant of proportional contoller.
KP = VO R =1 + F Vi RI

Where Vo = output voltage Vi = input voltage Rf = feedback resistance Ri = input resistance Kp = gain constant The control action of proportional plus integral controller is defined by the equation Vo (t) = Kp Vi (t) + Vi (t) (Kp / Ti) The transfer function of the controller is T.F = Vo (s) / Vi (s) = Kp (1 + 1 / sTi) Where Vo = output voltage Vi = input voltage Ti = integral time constant Kp = gain constant Both Kp and Ti are adjustable. The integral time constant Ti adjusts integral control action while changed in the values of Kp affects both the integral parts of the control action. The inverse of integral time Ti is called the reset rate. The reset rate is the number of time per minute that the proportional part of the control action is duplicated. Reset rate is measured in terms of reset per minute. The combinations of the proportional control action, integral control action and
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Control Systems Laboratory Manual

derivative combined action has the advantages of the three individual control actions. The transfer function of the controller is given by Vo (s) / Vi (s) = Kp (1 + 1 / (Ti + Td)) where Vo = output voltage Vi = input voltage Ti = integral time constant Td = derivative time constant Kp = gain constant PROCEDURE: PROPORTIONAL CONTROLLER: 1. Make the connections as per the circuit diagram 2. Set the values by using knobs on the trainer as follows: ♣ Input amplitude to 1 V (p-p) ♣ Frequency at low value 4. For various values of Kc, observe the waveforms PROPORTIONAL INTEGRAL CONTROLLER: 1. Make the connections as per the circuit diagram 2. Set the values by using knobs on trainer as follows ♣ Input amplitude to 1V (p-p) ♣ Frequency at low value and Ki to zero. 3. Keep Kc = 0.6 and increase Ki in small steps and observe the waveforms. PROPORTIONAL INTEGRAL DERIVATIVE CONTROLLER: 1. Make the connections as per the circuit diagrams. 2. Set the values by using knobs on trainer as following. ♣ Input amplitude to 1V (p-p) ♣ Kc = 0.6, Ki = 54.85, Kd = 0 ♣ Frequency at low values 3. The system shows fairly large overshoot.
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Control Systems Laboratory Manual

The above steps are repeated for a few non – zero values of Kd. 5. The improvement in transient performance is observed using CRO with increasing values of Kd, while the steady state error remains unchanged. 6. Calculate the values of peak overshoot and steady state error.
4.

Result:

DESIGN OF LAG AND LEAD COMPENSATORS EXPT.NO : DATE :

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Control Systems Laboratory Manual

APPARATUS REQUIRED: System employed with MATLAB 6.5 THEORY: The control systems are designed to perform specific taskes. When performance specification are given for single input. Single output linear time invariant systems. Then the system can be designed by using root locus or frequency response plots. The first step in design is the adjustment of gain to meet the desired specifications. In practical system. Adjustment of gain alone will not be sufficient to meet the given specifications. In many cases, increasing the gain may result poor stability or instability. In such case, it is necessary to introduce additional devices or component in the system to alter the behavior and to meet the desired specifications. Such a redesign or addition of a suitable device is called compensations. A device inserted into the system for the purpose or satisfying the specifications is called compensator. The compensator behavior introduces pole & zero in open loop transfer function to modify the performance of the system. The different types of electrical or electronic compensators used are lead compensator and lag compensator. In control systems compensation required in the following situations. 1. When the system is absolutely unstable then compensation is required to stabilize the system and to meet the desired performance. 2. When the system is stable. Compensation is provided to obtain the desired performance. LAG COMPENSATOR: A compensator having the characteristics of a lag network is called a lag compensator. If a sinusoidal signal is applied to a lag network, then in steady state the output will have a phase lag with respect input.

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Control Systems Laboratory Manual

Lag compensation result in a improvement in steady state performance but result in slower response due to reduced bandwidth. The attenuation due to the lag compensator will shift the gain crossover frequency to a lower frequency point where the phase margin is acceptable. Thus the lag compensator will reduce the bandwidth of the system and will result in slower transient response. Lag compensator is essentially a low pass filter and high frequency noise signals are attenuated. If the pore introduce by compensator is cancelled by a zero in the system, then lag compensator increase the order of the system by one. LEAD COMPENSATOR: A compensator having the characteristics of a lead network is called a lead compensator. If sinusoidal signal is applied to a lead network, then in steady state the output will have a phase lead with respect to input. The lead compensator increase the bandwidth, which improves the speed of response and also reduces the amount of overshoot. Lead compensation appreciably improves the transient response, whereas there is a small change in steady state accuracy. Generally lead compensation is provided to make an unstable system as a stable system. A lead compensator is basically a high pass filter and so it amplifies high frequency noise signals. If the pole is introduced by the compensator is not cancelled by a zero in the system, then lead compensator increases order of the system by one. FORMULA:
Gain = B y0 = A x0
= 20 log( B / A)

Phase θ = − sin −1 ( x0 / A) = − sin −1 ( y 0 / B )

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Control Systems Laboratory Manual

PROCEDURE: With out compensator:
1. Make the connection as per the circuit diagram.

2. Apply the 2V p-p sin wave input and observe the waveform.
3. Very the frequency of the sin wave input and tabulate the values of xo and yo

4. Set the amplifier gain at unity. 5. Insert the lag compensator with the help of passive components and determine the phase margin of the plant. 6. Observe the step response of the compensated system. PROCEDURE:
1. Enter the command window of MATLAB.

2. Create a New M-File by selecting file New M-File. 3. Type and save the program. 4. Execute the program by pressing F5 or Debug Run. 5. View the results. 6. Analyze the Results. With lead compensator:
1. Enter the command window of the MATLAB.

2. Create a new M – file by selecting File – New –M-File. 3. Type and save the program.
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Control Systems Laboratory Manual

4. Execute the program by either pressing F5 or Debug – Run. 5. View the results.
6. Analyze the results.

To determine the transfer function of the DC shunt motor APPARATUS REQUIRED: S.No Name of the Equipment Range Type Quantity

THEORY:
Speed can be controlled by varying (i) flux per pole (ii) resistance of armature circuit and (iii) applied voltage. It is known that N ∝ Eb. If applied voltage is kept, Eb = V – IaRa will

Remain constant. Then, N ∝ 1 φ By decreasing the flux speed can be increased and vice versa. Hence this method is called field control method. The flux of the DC shunt motor can be changed by changing field current, Ish with the help of shunt field rheostat. Since the Ish relatively small, the shunt filed rheostat has to carry only a small current, which means Ish2 R loss is small. This method is very efficient. In non-interpolar machines, speed can be increased by this methods up to the ratio 2: 1. In interpolar machine, a ratio of maximum to minimum speed of 6:1 which is fairly common. FORMULA: Armature Control D.C. Shunt motor: It is DC shunt motor designed to satisfy the requirements of the servomotor. The field excited by a constant DC supply. If the field current is constant then speed is directly proportional to armature voltage and torque is directly proportional to armature current.
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Field Control D.C. Shunt motor: It is DC shunt motor designed to satisfy the requirements of the servomotor. In this motor the armature is supplied with constant current or voltage. Torque is directly proportional to field flux controlling the field current controls the torque of the motor. K Transfer Function = Js2 (1 + ζ s)
   

PROCEDURE FOR TRANSFER FUNCTION OF ARMATURE CONTROL DC SHUNT MOTOR: Finding Kb 1. Keep all switches in OFF position. 2. Initially keep voltage adjustment POT in minimum potential position. 3. Initially keep armature and field voltage adjustment POT in minimum position. 4. Connect the module armature output A and AA to motor armature terminal A and AA respectively, and field F and FF to motor field terminal F and FF respectively. 5. Switch ON the power switch, S1, S2. 6. Set the field voltage 50% of the rated value. 7. Set the field current 50% of the rated value. 8. Tight the belt an take down the necessary readings for the table – 1 to find the value of Kb. 9. Plot the graph Torque as Armature current to find Kt.
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Control Systems Laboratory Manual

1. 2.

Finding Ra Keep all switches in OFF position. Initially keep voltage adjustment POT in minimum position. 3. Initially keep armature and field voltage adjustment POT in minimum potential position. 4. Connect module armature output A and AA to motor armature terminal A to AA respectively. 5. Switch ON the power switch and S1. 6. Now armature voltage and armature current are taken by varying the armature POT with in the rated armature current value. 7. The average resistance value in the table -2 gives the armature resistance.

PROCEDURE FOR TRANSFER FUNCTION OF FIELD CONTROL D.C. SHUNT MOTOR: Finding Rf 1. Keep all switches in OFF position. 2. Keep armature field voltage POT in minimum potential position. 3. Initially keep armature and field voltage adjustment POT in minimum potential position. 4. Connect module filed output F and FF to motor filed terminal F and FF respectively. 5. Switch ON the power, S1 and S2. 6. Now filed voltage and filed current are taken by varying the armature POT with in the rated armature current value. 7. Tabulate the value in the table no – 3 average resistance values give the fied resistance. Finding Zf 1. Keep all switches in OFF position. 2. Keep armature and field voltage POT in minimum position. 3. Initially keep armature and field voltage adjustment POT in minimum position. 4. Connect module varaic output P and N to motor filed terminal F and FF respectively. 5. Switch on the power note down reading for the various AC supply by adjusting varaic for the table no – 4.

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Control Systems Laboratory Manual

Finding Ktl 1. Keep all switches OFF position. 2. Initially keep voltage adjustment POT in minimum potential position. 3. Initially keep armature and field voltage adjustment POT in minimum position. 4. Connect the module armature output A and AA to motor armature terminal and AA respectively, and field F and FF to motor field terminal F and FF respectively. 5. Switch ON the power switch, S1 and S2. 6. Set the filed voltage at rated value (48V). 7. Adjust the armature voltage using POT on the armature side till it reaches the 1100 rpm. 8. Tight the belt and take down the necessary reading for the table – 5 Ktl 9. Plot the graph Torque as Field current to find Ktl OBSERVATION TABLE FOR TRANSFER FUNCTION OF ARMATURE CONTROL DC SERVO MOTOR: Table No:3 To find Rf Sl.No If (amp)